A strong approximation theorem for sums of random vectors in the domain of attraction to a stable law
The authors improve the a.s. invariance principle by \textit{W. Stout} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 49, 23-32 (1979; Zbl 0388.60035)] concerning the approximation of partial sums of i.i.d. symmetric random variables in the domain of attraction to a stable law of exponent \(\alpha\in (0,2)\). Moreover, instead of the quantile transform method employed by Stout, the multidimensional approximation result of \textit{J. Berkes} and \textit{W. Philipp} [Ann. Probab. 7, 29-54 (1979; Zbl 0392.60024)] is used. This allows for extensions to higher dimensions. In fact, the authors prove such an extension for \({\mathbb{R}}^ d\)-valued random variables, doing for \(\alpha\in (0,2)\), what the last author [Lect. Notes Math. 709, 171-193 (1979; Zbl 0418.60013), Theorem 2] did for \(\alpha =2\). A further extension to B-valued random variables appears to be possible.
scientific article; zbMATH DE number 3996753